Question

What is the LCM of 3x²y⁶ and 5x³y²?

Answer 1

The LCM is `15  x^3  y^6`

Explanation:

The LCM has to be exactly large enough to let every factor go into it evenly -- but no larger.

The coefficient is `15` because both `3` and `5` go into `15` evenly.

The power of `x` is `x^3` because both `x^2` and `x^3` go into `x^3` evenly.

The exponent for `y` is `y^6` because both `y^6` and `y^2` can be divided into `y^6` evenly.

Answer:
The LCM is `15 x^3 y^6`

`color(white)(mmmm)`―――――――――

When you are deciding which exponent you should use for `x`:

If you multiply the two exponents -- (`2 xx 3)`, you will get `x^6`.

It might be tempting to do that, because `x^6` can also be divided evenly by both `x^2` and `x^3`.

The problem is that, while `x^6` actually is a common multiple, it is not the least common multiple.
The LEAST common multiple is `x^3`.

This is similar to the decisions you make when you are finding a Least Common Denominator.

If you have to add `(1)/(2) + (1)/(6)`, it might be tempting to just multiply `2 xx 6` and use twelfths as the common denominator.

It actually is a common denominator, but not the LEAST one.
The least common denominator is `6` because both `2` and `6` will divide evenly into `6`.

`color(white)(mmmm)`―――――――――

On the other hand, sometimes it isn't worth the trouble to figure out the absolutely smallest possible common denominator -- or the absolutely least common multiple.

In that case, you can always just multiply the exponents (or denominators) and just go ahead and use that number.

It may not be the LEAST possible number, but it's better than nothing -- and better than wasting a lot of time if the very LEAST one isn't obvious.

Example:

Add
`(1)/(6) + (1)/(10) + (1)/(5)`

It might be easier -- and faster -- just to multiply all the denominators together and use `300` as the common denominator.

Or maybe you can see that you don't have to repeat the `5` (because it's already part of the `10` anyway), so you just get `60` as the common denominator.

The drawback is that `300` and `60` are large denominators, so using them makes the problem a little harder.

But on the other hand, it is easy enough to reduce fractions to their lowest terms later when you see a good chance.

And you can be sure that `300` (or `60`) will actually be a common denominator.

And finding those numbers is fast and easy -- maybe faster than figuring out that the very LEAST denominator that they have in common is `30.`

Answer 2

`LCM = 15x^3y^6`

Explanation:

The LCM (Lowest Common Multiple is the smallest expression which is divisible by both the terms given. It must contain all the factors of `3x^2y^6` and all the factors of `5x^3y^2`, but without any duplicates.

Write each term as the product of its prime factors:

`3x^2y^6 = color(blue)(3)" "xx x xx x" "xx color(blue)(yxxyxxyxxyxxyxxy)`
`5x^3y^2=ul( " "color(blue)(5)xx color(blue)(x xx x xx x) xxyxxycolor(white)(wwwwwwwwww))`
`LCM= color(blue)(3xx5xx x xx x xx x xxyxxyxxyxxyxxyxxy)`

`LCM = 15x^3y^6`